Coded Caching for Hierarchical Two-Layer Multi-Access Networks with Low Coding Delay

We consider the two-layer hierarchical coded caching networks, where users connected to each mirror are connected to multiple caches instead of having a dedicated cache. The problem setting involves a server with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> files connected to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K_{1}$</tex> mirrors, each with a cache size of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M_{1}$</tex> files, through an error-free bottleneck link, which are further connected to users each one of them can access <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$r$</tex> out of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C$</tex> caches each of size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M_{2}$</tex> files, available in the system. Each mirror is connected to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\binom{C}{r}$</tex>, users, and no two of these users connect to the same set of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$r$</tex> caches. The total cache size in the system is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\overline{M}=K_{1}M_{1}+K_{1}CM_{2}$</tex> called the global memory. The case <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$r=1$</tex> corresponds to hierarchical two-layer systems with users having dedicated caches. Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R_{1}$</tex> be the rate (in the unit of files) of the transmission from the server to the mirrors and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R_{2}$</tex> be the rate of transmission from any mirror to the users connected to it that satisfy the demands of all the users. The total delivery time from the server to a user is indicated by the metric <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R_{1}+R_{2}$</tex>, which is also known as the coding delay. In this work, we present an achievable scheme with low coding delay for a given global memory size. We compare <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R_{1}+R_{2}$</tex> vs. global memory <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\overline{M}=K_{1}M_{1}+K_{1}CM_{2}$</tex> of our scheme with other existing schemes for hierarchical two-layer coded caching networks. As we increase <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$r$</tex>, our scheme has a reduction in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$R_{1}+R_{2}$</tex>, which provides a dual advantage of serving a larger number of users while having a lower coding delay.